Combinatorics Seminar

When: Sunday, March 15, 10am Where: Schreiber 309 Speaker: Ehud Friedgut, Weizmann Institute Title: A sharp threshold for the collapse of the random triangular group.

Abstract:

Consider a random group with n generators, and a (random) collection of relations of size three. We study the evolution of such a group where the relations are added one by one at random, or (almost) equivalently, where each possible relation is chosen independently at random with probability p, and p grows from 0 to 1. We show the existence of c=c(n), a function bounded between two constants, such that if p=(1+o(1))cn^{-3/2} the group is asymptotically almost surely trivial, whereas if p=(1-o(1))cn^{-3/2} then the group is a.a.s. non-trivial. In fact, if p is just slightly smaller (I'll define "slightly" in the talk) then the group is a.a.s. infinite, torsion free and hyperbolic. In this talk we'll mainly study the geometry of the random group, but won't assume any prior knowledge regarding this notion.

Joint work with Sylwia Antoniuk and Tomasz Luczak.